In this paper an algorithm is presented for integrating expressions of the form $\smallint ge^f dx$, where f and g are rational functions of x, in terms of a class of special functions called the special incomplete $\Gamma $ functions. This class of special functions includes the exponential integral, the error function, the sine and cosine integrals, and the Fresnel integrals. The algorithm presented here is an improvement over those published previously for integrating with special functions in the following ways: (i) This algorithm combines all the above special functions into one algorithm, whereas previously they were treated separately. (ii) Previous algorithms require that the underlying field of constants be algebraically closed. This algorithm, however, works over any field of characteristic zero in which the basic field operations can be carried out. (iii) This algorithm does not rely on Risch’s solution of the differential equation $y' + fy = g$. Instead, a more direct method of undetermined coefficients is used.

This paper describes the design of an algebra system Views implemented in Smalltalk. Views contains facilities for dynamic creation and manipulation of computational domains, for viewing these domains as various categories such as groups, rings, or fields, and for expressing algorithms generically at the level of categories. The design of Views has resulted in the addition of some new abstractions to Smalltalk that are quite useful in their own right. Parameterized classes provide a means for run-time creation of new classes that exhibit generally very similar behavior, differing only in minor ways that can be described by different instantiations of certain parameters. Categories allow the abstraction of the common behavior of classes that derives from the class objects and operations satisfying certain laws independently of the implementation of those objects and operations. Views allow the run-time association of classes with categories (and of categories with other categories), facilitating the use of code written for categories with quite different interpretations of operations. Together, categories and views provide an additional mechanism for code sharing that is richer than both single and multiple inheritance. The paper gives algebraic as well as non-algebraic examples of the above-mentioned features.

Since R. Risch published an algorithm for calculating symbolic integrals of elementary functions in 1969 (Traps. Amer. Math. Soc., 139 (1969), pp. 167–189), there has been an interest in extending his methods to include nonelementary functions. In this paper, we use the framework of differential algebra to make precise the notion of integration in terms of elementary functions and logarithmic integrals. Basing our work on a recent extension of Liouville’s theorem on integration in finite terms, we then describe a decision procedure for determining if a given element in a transcendental elementary field has an integral which can be written in terms of elementary functions and logarithmic integrals. This algorithm first examines the structure of the integrand in order to limit the logarithmic integrals which could appear in the integral to a finite number. This allows us to write a general expression for the integral and then use techniques similar to those employed by Risch to calculate the undetermined parts.